Linear Algebra Vectors in the Plane

Welcome back to educator.com, this is a continuation of Linear Algebra, today we are going to be talking about vectors in the plane, so the plane is also represented as something called, R² which just means the real number squared.0000

Basically you have the X axis t, which is the real numbers, and we just take another, a copy of the real numbers and we make it perpendicular, which is why we call it R² by that analogy, normal space would be called R³0014

And n space is called Rⁿ, R raised to the n power.0027

Okay, let's go ahead and get started.0031

Okay, in math and science we talk about two types of quantities, one is a scalar, which is just a fancy word for a number and the other is something called a vector.0040

And in case you are wondering why it is that we actually differentiate, why would we even need something like a vector.0053

what I would like to tell my students is think of a pushing analogy.0059

If somebody comes, let's say in front of you an pushes you with a certain force, let's just say it's a 100 newtons of force, that's a number.0062

Well, you end up going backward in one direction, so let's say you are standing over here.0073

If they push you this way, you are moving, you are going to end up being pushed that way.0079

Let's say somebody comes from the other direction and pushes you in that direction, well you end up moving that way as it turns out, you end up in different places.0085

You end up moving in different direction, but they are both pushing with the same force, so there is a difference because this is not the same motion, so as it turns out in the real world.0094

We need something more than just a number, a particular situation needs to have, certain situations not all of them, need to have some other quality associated with them.0105

And that quality is a direction, so if I say i am going to push you with a 100 newton’s of force in this direction.0116

That is a, we call it a vector having a length of 10, I mean 100, what we call a magnitude in that direction.0124

If it were the other way, well we say it's a vector whose magnitude is still a 100 but it's in this direction.0133

And these are two very different vectors, because they have different direction, even though their magnitude is the same, so that sort of the unqualitated description of what a vector is.0140

Okay, let's go ahead and talk about a reference frame for vectors, so we take as our reference frame, the standard XY coordinate plane, the Cartesian plane and to the right of the X is positive and up on the Y axis is positive, negative, negative.0150

Nothing that you don't already know.0171

Now, if I start at the origin and if I draw, well let's not draw it, first let me just pick a point, so I have this point, let's say the point is (2, 4).0176

Well yes it's true, it represents a point in space, but if I start from the origin and put the tail of an error there, and if I go and put ahead of the error, something like that.0186

Notice, I have actually now given you an explicit direction from a particular point origin, well since this Cartesian coordinate plane is our frame of reference.0196

The origin will be our ultimate point of reference, so now I have an error associated with this, you know this coordinate here (2, 4).0207

Well, this coordinate is called a vector and this error is also called a vector, they are just two different representations of it, so if I call this vector U...0219

I can certainly represent it as (2, 4).0232

And another representation of it that I will also see is I will write it as a column matrix instead of a, I will write it like this (2,4), so notice this is two rows and one column.0235

And remember anything that has either one row or one column, we called it a vector, so now we can see why we can associate this idea of a vector with a matrix.0248

We don't necessarily have to have this coordinate with a comma in between, we can just represent a vector as a point to (2,4).0258

And again you are also welcome to write it as a row vector (2, 4), not necessarily with the coordinate, mean the only difference being that little comma there.0266

This is still saying move 2 in the X direction, 4 in the Y direction, 2 in the X direction, and 4 in the Y direction and now we have introduced this other notion of it actually being an error from the origin to this particular point.0275

With this error, now we have a physical something and now there's something else that we can associate with it.0290

We can associate a length with this error, because it has a particular length, and we can associate an angle from a reference line.0297

Well we take our reference line as the X axis and we measure all angles in counter clockwise direction.0305

Well, that would be considered a positive angle and if I go this way, this would be consider a negative angle.0312

If I go all the way around ones that 360 degree, if I go around twice, that is a 720 degrees, two times 360, yes it's 720.0318

Even though we end up in the same place, the angle measure is actually different, so again...0330

...Counter clockwise positive angle, clockwise negative angle.0338

X axis is our reference line, the origin is our reference point, okay, let's define a couple of things.0343

If we have a vector, let's just take a generic vector U, and again vectors have the little error on top of it, and I will do it as a column this time.0353

I will often do both and it's not really a problem, later on when we get into certain aspects of linear algebra, it's going to be important on how we actually use a vector, whether we do it as a column or a row, but for right now it's not really much of a problem.0365

We define the magnitude, the symbol for the magnitude is oops, excuse me...0379

U is our vector, we have the symbol for the vector, and we put two double lines around it, that's the magnitude and that's just the length.0388

Well, you know that if this is our vector, and let me actually draw it again over here so make it little more clear.0396

well if I have this particular thing and here is the point XY, well you know that we moved to the right X units and we have moved up Y units, so this is Y and this is X.0403

Well the Pythagorean theorem tells us that X, ² + Y² = this length², so we define the magnitude as X² + Y² under the √ sign.0416

That gives us the length of the vector, or...0430

...The magnitude...0436

...Of these fancy words, now we can also define angle, so if we call this angle θ0440

Well we have Y, we have X, the relationship between θ is tangent, so if I have the tangent of θ = Y 0ver X.0448

Well, that implies that the angle θ itself is going to be the arc tangent or the inverse tangent, of what oops, little lines.0463

I don't want them to get in the way here of what we do.0476

Y/X, so when you are given a vector in this for, you can find out the length, and you can find out the angle that it makes with the positive X axis and remember again we are measuring that way, okay.0481

Okay, let's do an example here, so let's say I have the vector is (3,), okay so 3 in the X direction, 7 in the Y direction, it's in the first quadrant, they are both positive.0505

The magnitude of U...0524

...Equals 3² + 7² under the √ sign 9 + 49 = 58, √ 58.0529

That is our magnitude, it is that long, and again you will find that when I come up with these numbers that are irrational under the square root sign, I often don't simplify.0541

I just leave them like that; it's not a problem at all.0551

Simplification often times believe it or not, things like reduction and simplification, I think it often obscure the mathematics, once you get a particular number, you are more than welcome to leave it like that.0555

The angle...0566

...is we said it's the arc tangent of Y/X, so 7/3, and we end up with 66.8 degrees.0570

Now notice we haven't drawn anything here, here we were talking about a physical object, we are talking about an angle which is a geometric notion, and we have expressed it in degrees.0581

You can express it in radians if you would like, if you remember 180 degrees is π radians, 3.14, you are welcome to do it either way, it's not a problem.0591

We have this geometric notion, we have a point that represents an error, you notice we haven't drawn any pictures here.0602

Now, you can certainly deal algebraically with vectors, it's not a problem, it's one of the reasons why we are doing it linear algebra.0609

Ultimately we want to take the geometric notion and bring it into the round of algebra, so we will give it a former foundation than just drawing pictures.0615

But, pictures are a big health and you want to make sure that you know what it is that you are dealing with, so we know that we are dealing with something in this quadrant .0622

When we get a number like 66.8 degrees, it actually makes sense, 3 this way, 7 this way, it actually should have been a little but steeper, I apologies.0631

But, again we want our numbers to make sense, so this is the number that is.....0640

....8 degrees and again we are measuring it from the positive X axis in that direction.0645

Okay, let's do another example, let's take a vector, let's actually draw this one out first, yeah let's draw it out over here, so I am going to have one of my vectors, so I give the names...0653

I will call one of them T, how's that for tail, and that's going to be (3,2), this time I wrote as a row vector and this other one I will call H for head, as in the error head.0675

And let's put this one at (7,4), okay so (3,2), oops (3, -2) actually, so it will go (1,2,3), (1,2).0691

This is the tail, that's one thing and the head is (7, 4), 4, 5, 6, 7 and we go up 1, 2, 3, 4 and we are over here, so we have this vector right here.0704

Now, notice this didn't begin at the origin, it's not a problem, any vector that we can move it to the origin, so when we actually try to find.0718

Well, think of it this way, if we just move this vector over to the origin, it's going to end up being something like this, so as it turns out, any vector in the plane whether it begins at the origin or not is in some equivalent to a vector that actually begins at the origin.0733

When we actually solve, for the magnitude and the angle θ, we are just going to be dealing in the same way we did before.0750

We just take the difference between these two, the difference from the head to the tail, so in this particular case our X value, in other words, this distance is just a difference between the X values here and here.0758

And our Y value for the vector is the difference between the Y values here and here, so let's find our X, it's equal to 7 -3, always do the head - tail, equals 4.0771

And our Y = 4 - (-2), which is 6, 4 - (-2), which is 6.0789

What we have is this thing here which doesn't begin at the origin is equivalent to a vector that does begin at the origin that has, well whose algebraic representation is (4, 6).0799

This vector is the (4,6) vector, well so is this, except it doesn't start at the origin, that's why it's represented by two different points, so for our practical purposes we can just deal with this one, they are equivalent, okay.0813

Let's do our, let's give this vector a name, let's just call it S.0828

The magnitude of S, again with double lines is equal to 4² + 6², under the √ sign 16 + 36 is 52, I hope if my arithmetic is correct.0839

I often make arithmetic mistakes, and again there is always going to be somebody there to check your arithmetic, there won't always be somebody to check your mathematics.0855

If you have to make a choice, mathematics comes first, not arithmetic.0863

And our θ is equal to the inverse tangent of Y/X, 6/4.0869

You don't have to reduce, 56.3 degrees, it's...0878

... That is this angle which is the same as this angle because they are equivalent, not a problem, so again when you are dealing with a vector that's been expressed with a head and a tail, somewhere else except at the origin, we just treated the same way.0889

You just take the coordinate for the head - the tai, coordinate for the head - the tai, and you end up with a vector as if it is starting at the origin.0903

Okay, let's do one more example here, this time we will do, our vector is (-6, -3) so again picture is worth a 1000 words.0917

We use pictures to help us understand what's going on, pictures are not proofs algebra is proof, so (-6,-3)...0935

...Were somewhere here, so we are looking at something like that, so just, so we know what we are dealing with.0947

We are dealing with something in the third quadrant, so when we gets our numbers, we want to make sure that the numbers match o0our geometric into vision, the picture.0953

Let's take the magnitude of U, well it is -6², which is 36, -3² which is 9, all under the √ sign, which gives us a √ 45, if I am not mistaken, and our angle θ.0962

Here is where it's going to get interesting, the inverse tangent and I apologize, I am a little older so I was actually taught as arc tangent, but inverse tangent is fine.0984

Y/X, -3/-6, when you enter this into a calculator, here is what you are going to get, 26.5 degrees just on the surface that doesn't make sense.0995

we are in the third quadrant, we said that angles are measured from the +X axis over to that way, so I know that my angle has to be more than 180 degrees lees than 270 degrees, 26.5, doesn't really jive with that.1014

Here's what's going on, if you remember from your trigonometry, whenever you take the inverse function, remember the graph of the tangent function...1028

Well, I too, I am not going to, let's not worry about the gap, let me just say whenever you take, use your calculator, the value that it is going to give you for your angle is going to be and angle between -90 and +90.1040

Okay, because the period of the tangent function is π so, what this value represents, remember you doing a tangent, so if you drop a perpendicular down to the X axis and it's always down to the X axis you never drop a perpendicular to the Y axis.1057

This angle right here is what is 26.5, remember you are taking the arch tangent of a distance, okay.1076

-3/-6, well here is your -3, here is your -6, it's just a distance over a distance.1084

that's what the calculator is calculation, so for all practical purposes is acting as if the angle is somewhere here, that's why it's important to know where are in the third quadrant.1093

Now that you, and when we formally decide to measure this angle, we take the 180 + the 26.5, so our actual θ is not 26.5, but based on our standard of this being our reference line, this is 26.5.1104

From here, the formal θ is 180 + 26.5, which is 206.5, positive.1124

Okay, so we need to differentiate that, so there is a couple of things that we need to aware of, which is pretty characteristic if you remember from working with trigonometry in angles.1139

You have to be aware of which quadrant you are working in and you also has to be aware of the science of trigonometric functions for, because the cosine is positive in the fourth quadrant.1147

The sine is positive in this quadrant and the tangent is positive in this quadrant, so we need the picture to help us to make sense of the numbers, okay.1158

let's see what we have got, vector addition and scalar multiplicatio0on, okay, so now that we have these things called vectors, we need to do things with them, and we can multiply them by scalars.1174

And we actually add vectors together.1189

Kind of the same ways, numbers, let's see what we have.1193

Vector addition, we will, vector U = let's do (U1, U2) and the vector V = (V1, V2), these are just the X and Y components of this vector.1202

Notice there are no errors over there, then we define the sum U + the, well, all you do is you add then component wise.1224

You add the X components of U with the X component of V, so it is U1 + V1, then in this case, you know what, I think I am going to go ahead and put commas just to, and then you have U2 + V2.1236

We can also write it as equivalent to U1 + V1, U2 + V2, this is the column representation, so all I have done is I have added the X components, added my Y components and now I have a new vector which is U + V.1252

Let's talk about what this looks like geometrically okay, I am going to put my U vector right there, and I am going to put my V vector, I'll make it, I will make it kind of short, okay.1275

I don't want to run out of room, all this means is that do U, then do V.1291

In other words, U is here, and then just do V, well V is a vector that goes in this direction and that's wrong, so just lay it on top with that, and you end up at that point.1298

Well, as it turns out as you remember, that point forms a parallelogram, that is the end, that's your beginning point.1312

This is your ending point, so this vector, once you put the head here, this vector is our U + V vector.1320

Again all we have done is we have done U first, and then we have done V, if you add three vectors, four vectors, five vectors, you just keep adding them and moving along and where you end up, that's where the head of the final vector goes, and they all begin at the origin.1332

This is U + V, okay.1348

Let's also do U - V, now U - V, well there is no such thing as vector subtraction, but really what you are doing is U + -V, well the -V is just the V, same length in the opposite direction.1353

This would be -V, so now when we do U - V, that means do U first and then go V distance in the opposite direction., so we do U first and the we go.1374

Well V goes this way, so -V is this way, so we go in the opposite direction, we go down that way and we end here.1391

That vector...1403

... Is U - V, and you have treated the same way, if it's U - V, well it's this entry - this entry that forms the X coordinate, this entry - this entry, that forms that coordinate.1408

And again all you are doing is you are going along the vector, U + V is do U first, then do V, wherever 6you end up, that's where the head of the final error is concerned.1423

This is the vector U + V, this is the vector U - V, this is the original U , this is the original V and this is the -V, okay.1432

Okay, now let's talk about scalar multiplication....1448

...Okay, once again we will let U = let's say XY, we can also write it as XY column matrix, so let U = that and A is a scalar, just a number.1460

Then, see A times U = well, A times X, A times Y or AX, AY written as a column.1488

All I am doing is taking this scalar and multiplying it by every entry in the actual vector itself, what this means geometrically is the following.1514

If this is my vector U, well whenever I multiply by a constant, all I am doing is expanding it if the constant is greater than 1, I am shrinking it, if it is less than 1, and I am pushing it to the opposite direction, if it is negative.1526

If I have this vector, let's say it's XY, and if I multiply it by, that means it take that vector and I increase its length by 5.1551

that means I have increased this X value by 5, I have increased the UY value by 5, if I multiply by 1/5, I shrink it down by a fifth.1560

If I multiply it by -5, that means it is the length of 5 but in the opposite direction, that's all that's happening pictorially, geometrically, okay.1569

Now let's see what we have got, so let's take U = 6 and -9, so (6, - 9), we will put it in the fourth quadrant, no, yes, fourth quadrant.1585

And V = (3, 4), which we would put it in the first quadrant, so let's do U + V, that's equal to, I am going to write this as a column matrix, so 6 + 3 is 9.1600

Okay, and -9 + 4...1623

... 6 + 3 is 9, -9 + 4 is -5, so our U + V is that vector, how about U - V, well we do 6 - 3, and we do -9 - 4.1631

6 - 3 is 3, -9 - 4 is - 13...1647

...Algebraic, now let's see what this actually looks like geometrically to get a sense of what's going on.1667

We want our intuition and our algebra to match, so 6 - 9; put's me somewhere down, say down here, okay.1675

This is my U and again we are not , we don't have to be exact here, you are welcome to (inaudible 2813) if you want, that's always nice.1687

And (3,4), may be somewhere up here, okay so this is U and this is V, U + V means do U first and then do V.1695

That puts us right there, this is U + V, (9 , -5), (9,-5) yeah seems about right, should keep this in the fourth quadrant,1716

While U + V, I am sorry U - V, we have U and then -V, which is down this direction.1729

It's actually off the page, so it's going to keep as (3,-13), yes looks pretty good, (3,-13) yes it jives, it's exactly right, so it's going to be a vector, it's going to be, and a little further down.1740

Yes, everything seems good but again when doing these, it's the algebra that matters, we use the pictures to help us understand what's happening in order to make sense of the algebra, not the other way round, okay.1756

Angle between two vectors, okay draw picture here.1775

Let's just take one vector randomly there and another vector randomly there, there is an angle between those vectors.1782

Let's call that angle θ and I notice this is not the same θ as the angle of one of the vectors which is from the positive X axis.1792

This has a θ, this angle also has a θ but we are talking about the angle actually between them, and as it turns out there is a beautiful formula that allows us to work with this.1799

Let's say this is U and let's say this is say is V, we have two vectors in the plane, as it turns out the cosine of the angle between them which we call θ is equal to the dot product of those two things U.V over...1813

...The product of the magnitudes of those two vectors, and again θ in this case is going to be greater than 0, less than 180.1836

When you actually work this out, you are going to get some angle from 0 to 180, an angle being 0, that means the vectors are pointed in the same direction; the angle between them is 0.1846

The angle is 180, that means you have vectors that are, that angle is 180, so if it goes past that, well the answer you are going to get is that angle, not that angle, okay that's all this means.1858

Let's do an example, so if we have U = let's say (2,5) and V = (-3, 6).1874

Well let's do the dot product, U.V and you remember it's the product of the X values + the products of Y values down the line.1889

Two times -3 is -6 + 5 times 6, +30, -6 + 30 = 24, so that's our dot product.1900

That's going to be our numerator, and now the magnitude of U is going to be 2² which is 4, 5², which is 25, and the √ sign is √ 29.1912

The magnitude of V is -3² is 9, 6² is 36.1930

That is equal to 45, ye √ 45, therefore our cosine of θ using our formula up here is equal to 24/√29 times √45, which is 0.664.1939

And when I take the inverse cosine of that, okay so θ when we go over here.1964

θ = the inverse cosine of 0.664, I get 48.4, write this a little more clearly, my apologies.1973

48.4 degrees, so that tells me that if I have U (2, 5), that's in this quadrant, (-3, 6), that's in the second quadrant.1992

I am dealing with an angle between them of 48.4 degrees.2006

That's really nice to be able to be just given the vector values and to be able to extract some geometric property that is not necessarily implied by anything.2009

These are just sort of numbers representing things, and yet here we are able to tell you what the angle between those vectors is.2022

This is very extraordinary, okay I see.2028

If U is perpendicular to V at right angles, then of ‘course θ = 90 degrees.2036

Well, what's the cosine of 90 degrees? 0, so that means that 0 = I use U and V here.2051

Let me...2065

...U.V over the magnitude of U times the magnitude of V, well I just multiply both sides by the magnitude of U and the magnitude of V and I end up with U.V = 0.2070

Here we go so, as it turns out if the two vectors are perpendicular to each other, the dot product is 0.2094

If the dot product of two vectors is 0, they are perpendicular to each other, we don't say perpendicular, and we actually use the word orthogonal.2104

U and V are...2117

...Orthogonal if and only if equivalents U.V = 0, so if I am given two vectors, i take the dot, if they are equal to 0, I know that they are perpendicular.2128

If I know that they are perpendicular, I know that the dot product is 0, and the reason we say orthogonal is perpendicular is when we move to higher dimensions and when we actually move later on for those of you to go on to Mathematics.2141

You'll speak of actual functions that are orthogonal and it's defined in the similar way.2153

This is the power of abstract mathematics is we start with the things that we know, two space, three space, pictures that we can deal with a, and we can generalize to all kinds of mathematical structures that share the same properties.2158

We need a more general language to deal with them, so we don't talk about perpendicular functions, we speak about orthogonal functions.2171

And so we might as well start now and start dealing with orthogonal vectors, okay.2179

Well, let's do something else here, what if we had θ = 0 and let's take U.U, okay.2184

Well the cosine of 0 is 1, so let's actually write out our formulas.2201

Cosine of this, so the cosine of 0 degrees = 1, well the cosine of θ = let's take U.U, let's just dot it with itself.2213

Just put it into our definition for the angle between two vectors, so in other words I have U and I have U ion top of it, the angle between them is 0, so let me see if I can extract some information from this.2228

The magnitude of U times magnitude of U, okay, multiply through and I end up with U.U = the magnitude of U.2243

This is just a number squared, and if I take the square of both sides, I end up with magnitude of U = U.U.2259

Now I have another way of actually finding the magnitude, what I can do is I can just take U dotted by itself, and then just take the square root of that number.2275

Very good, okay....2287

...We will talk about some, let's talk about some properties of the dot product and unit vectors, okay all these properties are going to be reasonably familiar because we have mentioned them before.2293

U.U is greater than 0 if U is not equal to 0, and U.U = 0 if and only if U = 0.2309

In other words if U is not the 0 vector, I will put it, if U is not the zero vector, then dot product is always going to give you a positive number.2328

B, U.V = V.U, so the dot product is commutative, C, U + V.W = U.W, notation...2341

... + V.W in other words the dot product itself is distributive and the final one C, times U.V = I can pull the U out.2365

U.C times V or I can just take, pull the C out and do U.V, again just some properties to manipulate vectors when you start to deal with them.2385

Okay, we are moving along very nicely here, let us define a unit vector.2402

Unit vector is a vector...2411

...Whose length is 1, that’s it...2418

...A vector worse length is 1, okay and if you are given so let's just say, so let X be any vector, the unit vector which I'll actually as X with a little unit written down below.2426

Is equal to 1 over the magnitude of X, times the vector itself.2449

In other words I take the vector and I divide each entry of that vector by the magnitude of that vector.2460

Think of it this way, if I have the number 15 and if I want to turn it into 1, I divide it by itself right.2467

Yes, i just divide by 15 and I get a 1, it's a way in taking that number and converting it to a 1.2474

Well vector, we are also dealing with direction, so I can't divide by a vector, that's not defined in mathematics, but I can divide by a number, so if I take the actual vector itself, all of the components.2481

And if I divide each of the components which are numbers by the magnitude, which is a number, I essentially just scale it down by its magnitude.2493

In other words I turn it into a vector of length 1 in any direction, because we are talking about any vector, okay.2504

Let's see we have two very important unit vectors.2514

Okay, we have the vector in the X direction which we symbolize as I and we have the vector in the Y direction which can symbolize as J.2530

In other words, there is a unit vector length 1 that way, that's called I, and there is a unit vector right here and that's called J.2549

Well, as it turns out we can express any vector in the plane by a linear combination of these two and what that means is the following.2564

Let's say I have a vector X and let's say it is, I'll write it in, I'll write it in multiple forms, (7,9), which is equivalent to (7,9).2577

I want to express this as a combination of these unit vectors, well a unit vector is just a vector 1, well if I multiply it with this value in the X direction.2590

That means moving this direction and the direction of the unit vector, that many units, so another expression for this would be 7 times the unit vector I.2600

That means move 7 units in the direction of I + 9 units in the direction of J, and remember these are vectors.2613

Vector addition just means do this one first, then do this one, all of this is saying is, well, you know this is 7 in the X direction, 9 in the Y direction.2623

Well, this is saying 7 in the I direction and, which is the X direction and 9 in the J direction, so I have just sort of combined vector additions, scalar multiplications and I have represented with this very unit vectors the I and the J.2634

And any vector in here can be represented as a linear combination, and a linear combination just means a sum.2649

That's 1, that's 2, move seven units to the right, move 9 units up.2656

If I might have something like -6I, -3J, that means move 6 units in the opposite direction of I that means this way.2662

And move 3 units down in the J direction that means this way, that put's it somewhere over here.2673

Any vector in R2 can be represented by a linear combination of these two vectors, you will discover later.2681

Any vector in let's say 13 space, well I need 13 unit vectors and that's I can represent any of those vectors by all the 13 little unit vectors in that particular coordinate frame.2690

We will talk about that little bit later.2704

Okay, let's finish off with an example here, let's say we have the vector X, which is (-2, -3), so that puts us in the fourth quadrant.2708

Okay so we want to find a unit vector in this direction, okay.2722

Well let's go out and find what the magnitude of X is first.2729

It's going to be -2² 4, -3² 9, under the √ sign = √13, so our unit vector in the direction of X is equal to 1 over √ 13 times...2735

...-2, -3, which is equal to, just multiply it through.2758

-2 over √13 and -3 over √13 that is my new vector.2765

Notice I have an X coordinate and a Y coordinate; I have divided it by the magnitude of the vector itself.2773

This vector has a length of 1, in other words if I would have found the magnitude of this vector, I would do -2 over √13² + -3 over √13² under the √ sign, I end up getting 1.2780

That's the whole idea, very important concept, the unit vector.2795

And again notice that I have left this √13 in the denominator, it's not a problem, and it’s perfectly valid mathematics.2800

Don't let anybody tell you otherwise.2806

Thank you for joining us here at educator.com, we will see you next time for linear algebra.2810